There is a sequel to "The Lady or the Tiger" called the "Discourager of Hesitancy" which promises to reveal the secret of the ending. I always encourage my students to read it after reading "The Lady or the Tiger." Stockton uses the same style and bag of tricks to deliver another riveting, suspenseful tale of indecision!
I think #5 makes an excellent observation. This story is of course so famous and still studied today precisely because of the conflict external to the story, as we are left with the ending that gives us no indication whatsoever about how the story will actually end. This, in a sense, is the true conflict of the story, as it forces us to work out the kind of ending that we ourselves would want and then justify it. We move from spectators in the arena to the position of the ruler himself.
While the above answers are correct, I wish to offer another point-of-view. Perhaps it is not the conflict within the story which is most poignant, but instead the conflict created within the reader. I know that when my students read the story, a lot of conflict (external-man v. man) erupts. The students argue about what comes out of the door and why.
I think that this is the conflict which is the most important. Stockton has been criticized for leaving the story as he did. Readers want closure- Stockton does not provide closure (internal conflict for the reader and critic).
The king has no conflict at all in this story. His rules are clear, as are the punishments. He is semi-barbaric and has no personal attachments to those who have been accused. It could be said that his method of "justice" sets up a conflict, but the king himself is not plagued with any real feelings about anything. The princess's lover also experiences no conflict in this story. He does not worry about overstepping and loving someone he should not, and when he is a prisoner he trusts implicitly in the princess to determine his fate. So the only conflicted character is the princess. She is not conflicted about loving someone from a lower social station; and she suffers no conflict when she discovers the secret of the doors, obviously breaking the law in doing so. She is, however, VERY conflicted regarding which door she wants her lover to open. The conflict is between two stong emotions in this semi-barbaric princess: love and jealousy.
One of the major conflicts is within the princess. She has to decide for herself whether she wants to tell her lover to choose the door where her lover would be killed or the one where he would end up with a woman she hated. It's a tough decision for her -- does she love the man so much that she would prefer to have him live even though it meant he would be with a woman she hated? Or is she selfish enough to see him die instead? It's a conflict within herself.